Existence Of Equilibrium Research Articles

Driving Forces of the Consumer Price Index During the Crises in the Eurozone: Heterogeneous Panel Approach

This paper examines key driving forces of inflationary pressures, taking into account supply and demand side determinants and actions of policy makers, during the pandemic and geopolitical crises in the Eurozone. Using heterogeneous nonstationary macro-panel models, especially the Mean Group (MG) and Pooled Mean Group (PMG) methods in the period 2020q1–2024q4, it is concluded that the dominant determination of inflationary pressures comes from the supply side. There is a long-run positive equilibrium relationship between the growth of energy prices and the growth of the consumer price index (CPI), and between the index representing supply bottlenecks (SBI) and the growth of CPI, while the relationship with the unemployment rate is insignificant. Also, the existence of a long-run equilibrium between the interest rate and CPI is homogeneous due to the unique monetary policy on a sample, and negative, indicating the efficiency of that policy. However, the speed of adjustment of individual economies is heterogeneous, and in the case of Greece and Ireland, insignificant. The heterogeneous or insignificant response of Eurozone member states, especially related to core-periphery asymmetry, refers to the vulnerability and structural weakness of the Eurozone economies, and the need for deeper integration.

Comment on “Salesforce Compensation with Inventory Considerations”: A Rejoinder

In this rejoinder, we address the concern about equilibrium existence in our previous work published in 2013. By making the simple adjustment of discretizing inventory ordering amounts to prevent inventory orders in arbitrarily small increments, we show that the results from that paper hold, thus maintaining the robustness of the insights obtained. This paper was accepted by Jeannette Song, operations management.

Relaxed Equilibria for Time-Inconsistent Markov Decision Processes

This paper considers an infinite-horizon Markov decision process (MDP) that allows for general nonexponential discount functions in both discrete and continuous time. Because of the inherent time inconsistency, we look for a randomized equilibrium policy (i.e., relaxed equilibrium) in an intrapersonal game between an agent’s current and future selves. When we modify the MDP by entropy regularization, a relaxed equilibrium is shown to exist by a nontrivial entropy estimate. As the degree of regularization diminishes, the entropy-regularized MDPs approximate the original MDP, which gives the general existence of a relaxed equilibrium in the limit by weak convergence arguments. As opposed to prior studies that consider only deterministic policies, our existence of an equilibrium does not require any convexity (or concavity) of the controlled transition probabilities and reward function. Interestingly, this benefit of considering randomized policies is unique to the time-inconsistent case.

An aggregative approach to pricing equilibrium among multi‐product firms with nested demand

AbstractI show equilibrium existence for a price‐setting game among multi‐product firms facing a nested logit/CES demand. As opposed to previous research I allow arbitrary firm/nest overlap, making the result relevant for applied work. Additionally, under easy‐to‐verify conditions, I show that there exist extreme equilibria, which are the most and least preferred by consumers, and provide an algorithm to find them. This allows researcher to numerically verify equilibrium uniqueness in applications, that is, if the extreme equilibria are equal to each other. As a by‐product, I show that inverting FOCs correctly identifies the marginal costs that rationalize observed prices.

Insider trading in discrete time Kyle games

AbstractWe present a new discrete time version of Kyle’s (Econometrica 53(6):1315–1335, 1985) classic model of insider trading, formulated as a generalised extensive form game. The model has three kinds of traders: an insider, random noise traders, and a market maker. The insider aims to exploit her informational advantage and maximise expected profits while the market maker observes the total order flow and sets prices accordingly. First, we show how the multi-period model with finitely many pure strategies can be reduced to a (static) social system in the sense of Debreu (Proc Natl Acad Sci 38(10):886–893, 1952) and prove the existence of a sequential Kyle equilibrium, following Kreps and Wilson (Econometrica 50(4):863–894, 1982). This works for any probability distribution with finite support of the noise trader’s demand and the true value, and for any finite information flow of the insider. In contrast to Kyle (1985) with normal distributions, equilibria exist in general only in mixed strategies and not in pure strategies. In the single-period model we establish bounds for the insider’s strategy in equilibrium. Finally, we prove the existence of an equilibrium for the game with a continuum of actions, by considering an approximating sequence of games with finitely many actions. Because of the lack of compactness of the set of measurable price functions, standard infinite-dimensional fixed point theorems are not applicable.

A Novel Nonlinear SAZIQHR Epidemic Transmission Model: Mathematical Modeling, Simulation, and Optimal Control

Abstract This study presents a seven-compartmental nonlinear epidemic model to explore the dynamics and control of the epidemic. The model extends the traditional susceptible-infected-recovered (SIR) framework by including additional compartments for aware, infodemic, hospitalized, and quarantined populations. It also incorporates three Holling type II saturated nonlinear incidence rates to better represent the disease transmission process. The "infodemic population" refers to those spreading misinformation about the disease, its spread, control, and treatment. The study examines disease dynamics in the absence of a vaccine and identifies two key equilibria: the disease-free equilibrium (DFE) and the endemic equilibrium (EE). It is found that the DFE is locally asymptotically stable when the basic reproduction number (R_0) is below one and becomes unstable when R_0 exceeds one. The model's behavior at R_0=1 is analyzed using center manifold theory, revealing a forward bifurcation. Additionally, the existence of a positive endemic equilibrium is confirmed for R_0>1, with no backward bifurcation observed when R_0≤1. The EE is also shown to be locally asymptotically stable when R_0 is greater than one. Furthermore, the study formulates and mathematically analyzes an optimal control problem. Finally, numerical simulations are presented to support the analytical results.

An immuno-epidemiological model with non-exponentially distributed disease stage on complex networks

Most of epidemic models assume that duration of the disease phase is distributed exponentially for the simplification of model formulation and analysis. Actually, the exponentially distributed assumption on the description of disease stages is hard to accurately approximate the interplay of drug concentration and viral load within host. In this article, we formulate an immuno-epidemiological epidemic model on complex networks, which is composed of ordinary differential equations and integral equations. The linkage of within- and between-host is connected by setting that the death caused by the disease is an increasing function in viral load within host. Mathematical analysis of the model includes the existence of the solution to the epidemiological model on complex networks, the existence and stability of equilibrium, which are completely determined by the basic reproduction number of the between-host system. Numerical analysis are shown that the non-exponentially distributions and the topology of networks have significant roles in the prediction of epidemic patterns.

Complex dynamics of an SIHR epidemic model with variable hospitalization rate depending on unoccupied hospital beds

In this paper, we propose an susceptible-infectious-hospitalized-recovered (SIHR) epidemic model with a nonlinear hospitalization rate depending on the number of unoccupied hospital beds. Note that the number of all hospital beds is used as a measure of all available medical resources. The basic reproduction number is calculated using the next-generation matrix method. We analyze the existence of endemic equilibria and discuss the global stability of the disease-free equilibrium. Existence and stability of endemic equilibria indicate possible occurrences of bifurcations. We confirm the appearance of backward bifurcation, saddle–node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation using normal form theory and central manifold theory. Numerical simulations show that the dynamic behavior of the model undergoes a transition from forward bifurcation to backward bifurcation and saddle–node bifurcation when the number of total hospital beds is reduced. Our findings suggest that when the number of total hospital beds falls below a threshold, backward bifurcation will occur, meaning that the disease cannot be eliminated even if the basic reproduction number is below unity. Therefore, the number of hospital beds should be increased beyond the bed threshold during an outbreak of a disease, which has important implications for disease control.

Plankton interaction model: Effect of prey refuge and harvesting

Abstract Harmful algal blooms are one of the major threats to aquatic ecosystem. Some phytoplankton species produce toxins during algal bloom and affect other aquatic species as well as human beings. Thus, for the conservation of aquatic habitat, it is much needed to control such phenomenon. In the present study, we propose a mathematical model of toxin-producing phytoplankton and zooplankton species, which follows the Holling Type III functional response. We consider the effect of prey refuge and harvesting on both the species. Boundedness of the proposed model, existence of equilibria, and their stability have been discussed analytically. We also discuss the optimal harvesting policy and existence of bionomic equilibrium. The numerical simulation has also been performed. We identify the control parameters that are responsible for the system dynamics of the model. The parameter prey refuge has a great impact on the dynamics of the model system. Higher value of prey refuge leads to the stable dynamics. Also, the growth rate of phytoplankton acts as a control parameter for the dynamics of the model. The higher value of growth rate of phytoplankton is responsible for oscillatory behavior.

Untangling the mobility dynamics in a patchy environment: An SEIR epidemic model with vertical transmission

Congenital Rubella, AIDS, and other human diseases may pass from mother to fetus through the placenta. On the other side, insects and plants often transmit vertically by their eggs/seeds. Here, we study an SEIR epidemic model with vertical transmission in a multi‐patch context to explore the spread of disease in a population divided into distinct regions. The local and global stability of disease‐free equilibrium for the n‐patch model system has been determined in terms of the basic reproduction number . Moreover, we discuss the existence and stability of endemic equilibrium for different cases in two symmetric regions. Our findings indicate that when is less than one, the disease‐free equilibrium is globally asymptotically stable, whereas when is greater than one, it is unstable. As a result, for , the model system has an unique endemic equilibrium that is always locally asymptotically stable. The dynamics of the proposed model system depending on the distribution of infected individuals across various regions have been studied. The findings suggest that the movements from high disease risk regions to low disease risks may increase the number of the infected population in the community. The value of could be enhanced by increasing the values of any vertical transmission parameter. However, the changes in the values of are relatively minor. Our findings also imply that while migrations of all individuals across symmetric regions do not leave any impact on the equilibrium state of the infected population, they do affect the trajectory of the solution to reach a steady‐state. The flow of people between the two symmetrical regions could not alter the endemicity.

Pattern dynamics in a predator–prey model with Smith growth function and prey refuge in predator poisoned environment

In this article, we investigate temporal as well as spatiotemporal dynamics of a predator–prey system where prey grows according to Smith growth function and prey refuges in predator poisoned environment. Self as well as cross diffusion are taken into consideration to describe spatial moment of the species which make the system more realistic. This study aims to investigate the role of prey refuge, toxin-carrying carcass and the impact of cross-diffusion on the spatial distribution of species densities and associated patterns. The existence of equilibria and their stability conditions for non-spatial system are derived. We discuss the occurrence of Hopf bifurcation and detect the stability of limit cycle by computing first Lyapunov number. Temporal dynamics of the system is studied by varying the significant parameters. Study reveals that both prey refuge and fatality rate of predator due to toxin-carrying carcass, initially makes the system stable for low refuge and fatality rate but predator population goes to extinction as those effects increase. Turing instability conditions are listed and different instability regions so formed are discussed. In Turing region, pattern selection with the help of amplitude equation via weakly nonlinear analysis are performed which also numerically supported. Intensity of refuge and fatality rate due to toxin carrying carcass result paradox with real ecological world due to the presence of cross diffusion. Various kind of patterns such as mixture of stripe and spot, spot, irregular chaotic patterns emerge due to Hopf and Turing instability. Particularly, spatiotemporal chaos is a controversial topic because of its significant consequences for population dynamics. The results of this work are expected to contribute to a better understanding the level of prey refuge, use of toxin-carrying carcass considering movement in population and effect of cross-diffusion.

Bifurcations and Homoclinic Orbits of a Model Consisting of Vegetation-Prey-Predator Populations.

This study provides a comprehensive analysis of the dynamics of a three-level vertical food chain model, specifically focusing on the interactions between vegetation, herbivores, and predators in a Snowshoe hare-Canadian lynx system. By simplifying the model through dimensional analysis, we determine conditions for equilibrium existence and identify various types of bifurcations, including Saddle-Node and Hopf bifurcations. Additionally, the study explores codimension-two bifurcations such as Bogdanov-Takens (BT) and zero-Hopf bifurcations. Coefficient formulas of normal forms are derived through the use of center manifold reduction and normal form theory. The study also presents an approximation of homoclinic orbits near a BT bifurcation of the system by computing explicit asymptotics based on regular perturbation methods. Utilizing the MATLAB package MATCONT, a family of limit cycles and their associated bifurcations are computed, including limit point cycles, period-doubling bifurcations, cusp points of cycles, fold-flip bifurcations, and various resonance bifurcations (R1, R2, R3, and R4). The biological implications of the findings are discussed in detail, highlighting how the identified bifurcations and dynamics can impact the population dynamics of vegetation, herbivores, and predators in real-world ecosystems. Numerical experiments validate the theoretical results and provide further support for the conclusions.

The existence of Walrasian equilibrium: infinitely many commodities, measure space of agents, and discontinuous preferences

The existence of Walrasian equilibrium: infinitely many commodities, measure space of agents, and discontinuous preferences

Dynamic analysis and optimal control of a mosquito-borne infectious disease model under the influence of biodiversity dilution effect

In this paper, biodiversity and a saturation treatment rate are introduced into a type of mosquito-borne infectious disease model with an incubation time delay. Through the dynamical analysis of the model, conditions for the existence and stability of disease-free and endemic equilibria are determined. The basic reproduction number of the model is calculated, and the condition for the existence of backward bifurcation is outlined. The study finds that under the dual influence of biodiversity and saturation treatment, the threshold characteristic of the basic reproduction number becomes invalidated. When R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$R_{0}$\\end{document} is less than 1, the model may exhibit four equilibrium states, with both the disease-free equilibrium and the endemic equilibrium being locally stable. In this scenario, whether the virus will become extinct depends on the initial conditions. The study also finds that when the basic reproduction number R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$R_{0}$\\end{document} is greater than 1, the stability of the model is influenced by the time delay, with Hopf bifurcation occurring at a specific time delay. In addition, another novel contribution of this paper is the formulation of an optimal control problem that takes into account the minimization of damage caused by humans to biodiversity. Based on the Pontryagin’s maximum principle, the specific characteristics of the optimal control measures are given, and the optimal strategy is derived by comparing five groups of control strategies. The optimal control results highlight the synergistic effect of multiple control measures with biodiversity. Under optimal control, a significant complementary effect between medical inputs and the dilution effect of biodiversity is evident. The findings imply that maintaining high biodiversity levels can decrease the demand for medical resources in mosquito-borne disease control efforts.

Dynamic environment UAV deployment algorithm based on potential game theory

AbstractTo address the issue of low coverage resulting from the challenge of acquiring the optimal deployment position in commonly used distributed deployment algorithms, this study presents a three‐dimensional deployment algorithm for Unmanned Aerial Vehicles (UAVs) based on potential games. First, a local mutually beneficial game model is designed to demonstrate the existence of exact potential games and Nash equilibrium. The Nash equilibrium solution corresponds to the maximum coverage. Next, drawing inspiration from exploration, a solution method called Exploration Spatial Adaptive Play is proposed. It utilizes the maximum utility function value from multiple step sizes in the exploration direction to update the action selection probability, thereby ensuring the optimal deployment position in each decision cycle. To address the issue of sensor position error, a method for processing sensor position errors is proposed. The simulation results demonstrate that the proposed distributed deployment algorithm achieves higher coverage compared to commonly used methods.

Equilibria in abstract economies with a continuum of agents with discontinuous and non-ordered preferences

This paper focus on the problem of the existence of an equilibrium in abstract economies and exchange economies. Spanning over the literature we have managed to extend and generalize some previous results. In particular, we generalize the main theorem of Yannelis (1987) on the existence of an equilibrium in an abstract economy with a continuum of agents, by allowing for discontinuous preferences. As a corollary of this result, we extend the finite agent Cournot–Nash equilibrium existence theorems with discontinuous preferences (e.g., Reny, 1999; Bareli and Meneghel, 2013; He and Yannelis, 2016; among others), to a continuum of agents. We also obtain an existence theorem for an abstract economy which allows for a convexifying effect on aggregation and nonconvex strategy and constraint sets. Furthermore, our new main theorem is used to prove the existence of a Walrasian equilibrium with a continuum of agents with discontinuous, non-ordered, interdependent and price-dependent preferences and thus extending the results of Aumman (1966) and Schmeidler (1969).

Existence of equilibrium in a dynamic supply chain game with vertical coordination, horizontal competition, and complementary goods

We consider supply chain competition and vertical coordination in a linear–quadratic differential game setting. In this setting, supply chains produce complementary goods and each of them includes a single manufacturer and a single retailer who coordinate their decisions through a revenue-sharing contract with a wholesale price and a fixed sales revenue share. We study a multiple leader-follower Stackelberg game where the manufacturers are the leaders and the retailers are the followers. Competition occurs at both levels of the supply chains. Retailers play Nash and compete in price; manufacturers also play Nash but they compete in choosing their production capacities by exploiting the equilibrium price decisions made by the retailers. We show that open-loop Nash equilibria exist when the manufacturers only receive a wholesale price (there are no longer exploiting the equilibrium price decision made by the retailers, however). When the manufacturers receive both a wholesale-price and a share of the retailers’ sales revenues, equilibria generally no longer exist. The non-existence of an equilibrium stems from the fact that the manufacturers’ instant profits are discontinuous functions of their production capacities. This discontinuity leads to a major technical difficulty in that one cannot apply standard optimal control approaches to study the equilibria of the dynamic game. Our results illustrate the possibility that competition between supply chains might not be sustainable when they sell complementary products and rely on a revenue-sharing agreement.

Competitive networked bi-virus spread: Existence of coexistence equilibria

The paper studies multi-competitive continuous-time epidemic processes. We consider the setting where two viruses are simultaneously prevalent, and the spread occurs due to individual-to-individual interaction. In such a setting, an individual is either not affected by any of the viruses, or infected by one and exactly one of the two viruses. One of the equilibrium points is the coexistence equilibrium, i.e., multiple viruses simultaneously infect separate fractions of the population. We provide a sufficient condition for the existence of a coexistence equilibrium. We identify a condition such that for certain pairs of spread matrices either every coexistence equilibrium lies on a line that is locally exponentially attractive, or there is no coexistence equilibrium. We then provide a condition that, for certain pairs of spread matrices, rules out the possibility of the existence of a coexistence equilibrium, and, as a consequence, establishes global asymptotic convergence to the endemic equilibrium of the dominant virus. Finally, we provide a mitigation strategy that employs one virus to ensure that the other virus is eradicated. The theoretical results are illustrated using simulations.

Comment on “Salesforce Compensation with Inventory Considerations”

This note critically analyzes the conditions for equilibrium existence in a model involving incentive schemes and inventory management. It identifies a technical flaw where equilibrium is asserted to exist under certain parameter values when, in fact, it does not. However, by introducing a reasonable assumption that inventory can only be ordered in sufficiently large increments, the qualitative outcomes and key insights remain intact. This paper was accepted by Jeannette Song, operations management. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2023.03914 .

CAPM in Real World: Risk-Friendly Investments

The aim of this paper is to show how the existence of equilibrium in CAPM may be obtained when individuals/investors are risk friendly. This assumption is closer to the real world, since risk aversion is rare and the portfolios implying a greater payoff are the ones which increase the variance of the payoff itself. Specifically, we assume that the wage vectors of the individuals/investors do not lie in the market space, which is the usual assumption for CAPM, namely, the payoff of any portfolio is replicated by the wage of the individuals/investors. The interpretation of such an assumption is that the wealth of the individuals/investors may not invest in financial markets as a whole. The second main result is that the perception of risk by the objective probabilities for the states of the world does not affect the equilibrium existence. This is a consequence of the No-Arbitrage assumption. This condition may be stated in terms of the existence of the objective probabilities for the states of the world. Since CAPM is usually considered as a way to compare the return of the so-called “market portfolio” and the return of any other portfolio, in the final section of this paper, we do provide the regression form of this altered form of CAPM.